Question 1

Use inverse matrices to solve the system of linear equations. ​
​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)    B

Question 2

CD = DC.
​

Accepted Answer

A) True 
 B)False  True

Question 3

Suppose that the economy of a small nation has an electronics industry, a steel industry, and an auto industry, with the following technology matrix. ​
​   ​
If the nation wishes to have surpluses of 104 units of electronics production, 271 units of steel production, and 200 automobiles, find the gross production of each industry.
​

Accepted Answer

A)  electronics = 1254, steel = 1262, autos = 722 
B)  electronics = 1252, steel = 1262, autos = 724 
C)  electronics = 1254, steel = 1269, autos = 722 
D)  electronics = 1254, steel = 1269, autos = 724 
E)  electronics = 1252, steel = 1262, autos = 722 
A)  electronics = 1254, steel = 1262, autos = 722 
B)  electronics = 1252, steel = 1262, autos = 724 
C)  electronics = 1254, steel = 1269, autos = 722 
D)  electronics = 1254, steel = 1269, autos = 724 
E)  electronics = 1252, steel = 1262, autos = 722 E

Question 4

The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) ​   ​
How many units of fuels were required to produce 1,000 units of manufactured goods?
​

Accepted Answer

A)  12.1 
B)  8 
C)  0.8 
D)  3.3 
E)  33 
A)  12.1 
B)  8 
C)  0.8 
D)  3.3 
E)  33

Question 5

Consider the original budget matrix. Assume there is a 20% increase in manufacturing, a 4% increase in office, a 6% increase in sales, a 21% increase in shipping, a 5% increase in accounting, and a 4% decrease in management. Find the new budget matrix by developing a matrix A to represent these departmental increases and then computing the matrix BA.

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 6

Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y. ​

Accepted Answer

A)  ​  
​ 
B)  ​​   
C)  ​   ​
​​ 
D)  ​   ​
​​ 
E)  ​   ​
​ 
A)  ​  
​ 
B)  ​​   
C)  ​   ​
​​ 
D)  ​   ​
​​ 
E)  ​   ​
​

Question 7

The matrix that follows represents work efficiency for workers at nine different work centers. Suppose that the work centers changes in rules cause efficiency to decrease by 0.04 at work centers 2 and 5 and to decrease by 0.01 at work center 7. Write the matrix that describes the new efficiencies.​ ​
    ​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 8

What is the order of matrices A, B, C, D, E? ​
​   ,   ,   ,   ,   ​

Accepted Answer

A)  Matrix A is 2×2, matrix B is 3×2, matrix C is 2×3, matrix D is 3×3, matrix E is 5×2. 
B)  Matrix A is 3×1, matrix B is 3×3, matrix C is 2×5, matrix D is 12×11, matrix E is 3×7. 
C)  Matrix A is 2×2, matrix B is 2×3, matrix C is 3×2, matrix D is 3×3, matrix E is 2×5. 
D)  Matrix A is 2×2, matrix B is 3×3, matrix C is 3×3, matrix D is 3×3, matrix E is 5×5. 
E)  Matrix A is 1×2, matrix B is 3×2, matrix C is 2×3, matrix D is 12×3, matrix E is 3×2. 
A)  Matrix A is 2×2, matrix B is 3×2, matrix C is 2×3, matrix D is 3×3, matrix E is 5×2. 
B)  Matrix A is 3×1, matrix B is 3×3, matrix C is 2×5, matrix D is 12×11, matrix E is 3×7. 
C)  Matrix A is 2×2, matrix B is 2×3, matrix C is 3×2, matrix D is 3×3, matrix E is 2×5. 
D)  Matrix A is 2×2, matrix B is 3×3, matrix C is 3×3, matrix D is 3×3, matrix E is 5×5. 
E)  Matrix A is 1×2, matrix B is 3×2, matrix C is 2×3, matrix D is 12×3, matrix E is 3×2.

Question 9

A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

Accepted Answer

A)    ,   ,   , for any real number z. 
B)    ,   ,   
C)    ,   ,   
D)    ,   ,   
E)  no solution 
A)    ,   ,   , for any real number z. 
B)    ,   ,   
C)    ,   ,   
D)    ,   ,   
E)  no solution

Question 10

A clothing manufacturer has factories in Atlanta, Chicago, and New York. Sales (in thousands) during the first quarter are summarized in the matrix below. ​   ​
During this period, the selling price of a coat was $125, of a shirt $50, of a pair of pants $50, and of a tie $25. Use matrix multiplication to find the total revenue received by each factory.
​

Accepted Answer

A)  Atlanta $13,100,000; Chicago $11,300,000; New York $6,750,000; 
B)  Atlanta $9,725,000; Chicago $11,825,000; New York $4,950,000; 
C)  Atlanta $13,100; Chicago $11,300; New York $6,750; 
D)  Atlanta $9,725; Chicago $11,825; New York $4,950; 
E)  Atlanta $975,000; Chicago $1,157,500; New York $495,000. 
A)  Atlanta $13,100,000; Chicago $11,300,000; New York $6,750,000; 
B)  Atlanta $9,725,000; Chicago $11,825,000; New York $4,950,000; 
C)  Atlanta $13,100; Chicago $11,300; New York $6,750; 
D)  Atlanta $9,725; Chicago $11,825; New York $4,950; 
E)  Atlanta $975,000; Chicago $1,157,500; New York $495,000.

Question 11

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Which breed had the greatest improvement in rank?

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

We have encoded messages by assigning the letters of the alphabet the numbers 1-26, with a blank assigned the number 27, and by using an encoding matrix A that converts the numbers to the coded message. To decode any message encoded by A, we must find the inverse of A, denoted by A-1, and multiply A-1 times the coded message. Use this code and the given encoding matrix A to find the matrix that can be used to decode the given message. ​
The code matrix is   . The message is 33, 132, 214, 22, 110, 203, 34, 132, 201.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 13

Write the transpose of matrix A. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 14

If matrix M has element   , what is   ? ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 15

The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Let A represent the matrix that contains the number of each of these species in the United States in 2003 and B represent the matrix that contains the number of each of these species outside the United States in 2003. Find the matrix B - A. What do the negative entries in matrix B - A mean? ​
​

Accepted Answer

A)  ​  
The negative entries tell us that that particular species is extinct. 
B)  ​  
The negative entries tell us how many more endangered or threatened species are in foreign countries. 
C)  ​  
The negative entries tell us that there are more endangered or threatened species in foreign countries. 
D)  ​  
The negative entries tell us that that particular species is extinct. 
E)  ​  
The negative entries tell us how many more endangered or threatened species are in the United States. 
A)  ​  
The negative entries tell us that that particular species is extinct. 
B)  ​  
The negative entries tell us how many more endangered or threatened species are in foreign countries. 
C)  ​  
The negative entries tell us that there are more endangered or threatened species in foreign countries. 
D)  ​  
The negative entries tell us that that particular species is extinct. 
E)  ​  
The negative entries tell us how many more endangered or threatened species are in the United States.

Question 16

The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system. ​   ​

Accepted Answer

A)    ,   ,   
B)    ,   ,   
C)    ,   ,   
D)    ,   ,   
E)    ,   ,   
A)    ,   ,   
B)    ,   ,   
C)    ,   ,   
D)    ,   ,   
E)    ,   ,

Question 17

A closed model for an economy identifies government, the profit sector, the nonprofit sector, and households as its industries. Each unit of government output requires 0.3 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.4 unit of households input. Each unit of profit sector output requires 0.2 unit of government input, 0.3 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.4 unit of households input. Each unit of nonprofit sector output requires 0.2 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.6 unit of households input. Each unit of households output requires 0.05 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.25 unit of households input. Write the augmented matrix for this closed model of the economy. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 18

Use row operations on the augmented matrix to solve the given system of linear equations. ​   ​

Accepted Answer

A)    ,   ,   
B)    ,   ,   
C)    ,   ,   
D)    ,   ,   
E)    ,   ,   
A)    ,   ,   
B)    ,   ,   
C)    ,   ,   
D)    ,   ,   
E)    ,   ,

Question 19

The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) ​   ​
Which is most dependent on the service industry?
​

Accepted Answer

A)  manufacturing 
B)  utilities 
C)  service 
D)  fuels 
E)  agriculture and food 
A)  manufacturing 
B)  utilities 
C)  service 
D)  fuels 
E)  agriculture and food

Question 20

We have encoded messages by assigning the letters of the alphabet the numbers 1-26, with a blank assigned the number 27, and by using an encoding matrix A that converts the numbers to the coded message. To decode any message encoded by A, we must find the inverse of A, denoted by A-1, and multiply A-1 times the coded message. Use this code and the given encoding matrix A to find the matrix that can be used to decode the given message. ​
Code matrix is
​   The message is
​
138, 59, 72, 29, 143, 58, 57, 24, 101, 44
​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Use inverse matrices to solve the system of linear equations.

CD = DC.

Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y.

The matrix that follows represents work efficiency for workers at nine different work centers. Suppose that the work centers changes in rules cause efficiency to decrease by 0.04 at work centers 2 and 5 and to decrease by 0.01 at work center 7. Write the matrix that describes the new efficiencies.

What is the order of matrices A, B, C, D, E? , , , ,

A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Which breed had the greatest improvement in rank?

Write the transpose of matrix A.

If matrix M has element , what is ?

The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system.

Use row operations on the augmented matrix to solve the given system of linear equations.

The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) Which is most dependent on the service industry?

Linear Equations and Functions

Quadratic and Other Special Functions

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability and Data Description

Derivatives

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Exam 3: Matrices

Use inverse matrices to solve the system of linear equations. ​ ​ ​

CD = DC. ​

Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y. ​

What is the order of matrices A, B, C, D, E? ​ ​ , , , , ​

A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Which breed had the greatest improvement in rank?

Write the transpose of matrix A. ​ ​

If matrix M has element , what is ? ​ ​

The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system. ​ ​

Use row operations on the augmented matrix to solve the given system of linear equations. ​ ​

The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) ​ ​ Which is most dependent on the service industry? ​

Linear Equations and Functions

Quadratic and Other Special Functions

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability and Data Description

Derivatives

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Use inverse matrices to solve the system of linear equations.

CD = DC.

Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y.

What is the order of matrices A, B, C, D, E? , , , ,

Write the transpose of matrix A.

If matrix M has element , what is ?

The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system.

Use row operations on the augmented matrix to solve the given system of linear equations.

The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) Which is most dependent on the service industry?